Workshop on Innovative Time Integration

s of Talks In alphabetical order SECOND ORDER CONVERGENCE OF A LIE GROUP TIME INTEGRATION METHOD FOR CONSTRAINED MECHANICAL SYSTEMS Martin Arnold MARTIN LUTHER UNIVERSITY HALLE-WITTENBERG Classical ODE and DAE time integration methods have been applied successfully for more than two decades to constrained systems in technical mechanics. For large scale systems, Newmark type integrators are considered to be an interesting alternative in terms of numerical effort and stability. In the present paper we study the time integration of constrained systems M(q)q̈ =−g(q, q̇, t)−B>(q)λ , Φ(q) = 0 by a generalized-α time integration method qn+1 = qn +hq̇n +(0.5−β )han +βhan+1 q̇n+1 = q̇n +(1− γ)han + γhan+1 with (1−αm)an+1 +αman = (1−α f )q̈n+1 +α f q̈n , M(qn)q̈n =−g(qn, q̇n, tn)−B(qn)λ n , Φ(qn) = 0 and appropriate parameters α f , αm, β , γ , see [1] and the recent extension to constrained systems in a Lie group setting [2]. The order condition γ = 0.5+α f −αm guarantees second order convergence in the classical case as well as for the Lie group integrator [3]. Fig. 1: Spurious transient oscillations: Generalized-α method applied to benchmark Heavy top. A potential drawback of the methods are spurious transient oscillations of the constrained forces that result from an order reduction in the transient phase, see Fig. 1. We present a strict mathematical analysis of this phenomenon and show how to avoid the spurious oscillations by perturbed starting values q̇0, a0.